Quotient of sum of exps

Percentage Accurate: 99.0% → 98.5%

Time: 5.4s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. a = -2.00000000000000007e154
  2. b = -2.00000000000000007e154

Specification

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))

C

double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(a) + exp(b))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

Python

def code(a, b):
	return math.exp(a) / (math.exp(a) + math.exp(b))

Julia

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))

C

double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(a) + exp(b))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

Python

def code(a, b):
	return math.exp(a) / (math.exp(a) + math.exp(b))

Julia

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-128) (/ (exp a) (+ (exp a) 1.0)) (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-128) {
		tmp = exp(a) / (exp(a) + 1.0);
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 5d-128) then
        tmp = exp(a) / (exp(a) + 1.0d0)
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-128) {
		tmp = Math.exp(a) / (Math.exp(a) + 1.0);
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-128:
		tmp = math.exp(a) / (math.exp(a) + 1.0)
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-128)
		tmp = Float64(exp(a) / Float64(exp(a) + 1.0));
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 5e-128)
		tmp = exp(a) / (exp(a) + 1.0);
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 5e-128], N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 5.0000000000000001e-128

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 100.0%

      \[\leadsto \color{blue}{\frac{e^{a}}{1 + e^{a}}} \]

   if 5.0000000000000001e-128 < (exp.f64 a)

   1. Initial program 98.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 98.6%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-128}:\\ \;\;\;\;\frac{e^{a}}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{a}}{e^{a} + e^{b}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ (exp a) (+ (exp a) (exp b))))

C

double code(double a, double b) {
	return exp(a) / (exp(a) + exp(b));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = exp(a) / (exp(a) + exp(b))
end function

Java

public static double code(double a, double b) {
	return Math.exp(a) / (Math.exp(a) + Math.exp(b));
}

Python

def code(a, b):
	return math.exp(a) / (math.exp(a) + math.exp(b))

Julia

function code(a, b)
	return Float64(exp(a) / Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = exp(a) / (exp(a) + exp(b));
end

Wolfram

code[a_, b_] := N[(N[Exp[a], $MachinePrecision] / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing

3. Final simplification 98.8%

   \[\leadsto \frac{e^{a}}{e^{a} + e^{b}} \]
4. Add Preprocessing

Alternative 3: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -700:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= a -700.0) 0.0 (/ 1.0 (+ (exp b) 1.0))))

C

double code(double a, double b) {
	double tmp;
	if (a <= -700.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (exp(b) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-700.0d0)) then
        tmp = 0.0d0
    else
        tmp = 1.0d0 / (exp(b) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (a <= -700.0) {
		tmp = 0.0;
	} else {
		tmp = 1.0 / (Math.exp(b) + 1.0);
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if a <= -700.0:
		tmp = 0.0
	else:
		tmp = 1.0 / (math.exp(b) + 1.0)
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (a <= -700.0)
		tmp = 0.0;
	else
		tmp = Float64(1.0 / Float64(exp(b) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -700.0)
		tmp = 0.0;
	else
		tmp = 1.0 / (exp(b) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[a, -700.0], 0.0, N[(1.0 / N[(N[Exp[b], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -700

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 45.2%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   4. Step-by-step derivation

      1. +-commutative 45.2%

         \[\leadsto \frac{1}{\color{blue}{e^{b} + 1}} \]

      2. flip-+ 0.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{b} \cdot e^{b} - 1 \cdot 1}{e^{b} - 1}}} \]

      3. pow2 0.2%

         \[\leadsto \frac{1}{\frac{\color{blue}{{\left(e^{b}\right)}^{2}} - 1 \cdot 1}{e^{b} - 1}} \]

      4. metadata-eval 0.2%

         \[\leadsto \frac{1}{\frac{{\left(e^{b}\right)}^{2} - \color{blue}{1}}{e^{b} - 1}} \]

      5. expm1-undefine 1.0%

         \[\leadsto \frac{1}{\frac{{\left(e^{b}\right)}^{2} - 1}{\color{blue}{\mathsf{expm1}\left(b\right)}}} \]

   5. Applied egg-rr 1.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{{\left(e^{b}\right)}^{2} - 1}{\mathsf{expm1}\left(b\right)}}} \]
   6. Step-by-step derivation

      1. unpow2 1.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{b} \cdot e^{b}} - 1}{\mathsf{expm1}\left(b\right)}} \]

      2. prod-exp 1.0%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{b + b}} - 1}{\mathsf{expm1}\left(b\right)}} \]

      3. expm1-define 1.8%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{expm1}\left(b + b\right)}}{\mathsf{expm1}\left(b\right)}} \]

   7. Simplified 1.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 1.8%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}} \cdot \sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}}} \]

      2. sqrt-unprod 1.8%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)} \cdot \frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}}} \]

      3. pow2 1.8%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\left(\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}}} \]

      4. flip-+ 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\color{blue}{\frac{b \cdot b - b \cdot b}{b - b}}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

      5. +-inverses 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{\color{blue}{0}}{b - b}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

      6. +-inverses 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{0}{\color{blue}{0}}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

   9. Applied egg-rr 0.0%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{0}{0}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}}} \]

   11. Simplified 45.2%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{e^{b + b}}}} \]

   12. Taylor expanded in b around 0 3.1%

       \[\leadsto \color{blue}{1} \]

   14. Simplified 100.0%

       \[\leadsto \color{blue}{0} \]

   if -700 < a

   1. Initial program 98.4%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 98.1%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -700:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{b} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 20.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-40}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -2.7e-6) 1.0 (if (<= b 3e-40) (+ 0.5 (* a 0.25)) 0.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.7e-6) {
		tmp = 1.0;
	} else if (b <= 3e-40) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-2.7d-6)) then
        tmp = 1.0d0
    else if (b <= 3d-40) then
        tmp = 0.5d0 + (a * 0.25d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -2.7e-6) {
		tmp = 1.0;
	} else if (b <= 3e-40) {
		tmp = 0.5 + (a * 0.25);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -2.7e-6:
		tmp = 1.0
	elif b <= 3e-40:
		tmp = 0.5 + (a * 0.25)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -2.7e-6)
		tmp = 1.0;
	elseif (b <= 3e-40)
		tmp = Float64(0.5 + Float64(a * 0.25));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -2.7e-6)
		tmp = 1.0;
	elseif (b <= 3e-40)
		tmp = 0.5 + (a * 0.25);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -2.7e-6], 1.0, If[LessEqual[b, 3e-40], N[(0.5 + N[(a * 0.25), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if b < -2.69999999999999998e-6

   1. Initial program 95.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 93.6%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   4. Step-by-step derivation

      1. +-commutative 93.6%

         \[\leadsto \frac{1}{\color{blue}{e^{b} + 1}} \]

      2. flip-+ 93.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{b} \cdot e^{b} - 1 \cdot 1}{e^{b} - 1}}} \]

      3. pow2 93.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{{\left(e^{b}\right)}^{2}} - 1 \cdot 1}{e^{b} - 1}} \]

      4. metadata-eval 93.4%

         \[\leadsto \frac{1}{\frac{{\left(e^{b}\right)}^{2} - \color{blue}{1}}{e^{b} - 1}} \]

      5. expm1-undefine 93.4%

         \[\leadsto \frac{1}{\frac{{\left(e^{b}\right)}^{2} - 1}{\color{blue}{\mathsf{expm1}\left(b\right)}}} \]

   5. Applied egg-rr 93.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{{\left(e^{b}\right)}^{2} - 1}{\mathsf{expm1}\left(b\right)}}} \]
   6. Step-by-step derivation

      1. unpow2 93.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{b} \cdot e^{b}} - 1}{\mathsf{expm1}\left(b\right)}} \]

      2. prod-exp 93.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{b + b}} - 1}{\mathsf{expm1}\left(b\right)}} \]

      3. expm1-define 93.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{expm1}\left(b + b\right)}}{\mathsf{expm1}\left(b\right)}} \]

   7. Simplified 93.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 93.6%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}} \cdot \sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}}} \]

      2. sqrt-unprod 93.6%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)} \cdot \frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}}} \]

      3. pow2 93.6%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\left(\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}}} \]

      4. flip-+ 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\color{blue}{\frac{b \cdot b - b \cdot b}{b - b}}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

      5. +-inverses 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{\color{blue}{0}}{b - b}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

      6. +-inverses 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{0}{\color{blue}{0}}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

   9. Applied egg-rr 0.0%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{0}{0}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}}} \]

   11. Simplified 4.9%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{e^{b + b}}}} \]

   12. Taylor expanded in b around 0 91.7%

       \[\leadsto \color{blue}{1} \]

   if -2.69999999999999998e-6 < b < 3.0000000000000002e-40

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 99.8%

      \[\leadsto \color{blue}{\frac{e^{a}}{1 + e^{a}}} \]

   4. Taylor expanded in a around 0 74.8%

      \[\leadsto \color{blue}{0.5 + 0.25 \cdot a} \]
   5. Step-by-step derivation

      1. *-commutative 74.8%

         \[\leadsto 0.5 + \color{blue}{a \cdot 0.25} \]

   6. Simplified 74.8%

      \[\leadsto \color{blue}{0.5 + a \cdot 0.25} \]

   if 3.0000000000000002e-40 < b

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 91.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   4. Step-by-step derivation

      1. +-commutative 91.0%

         \[\leadsto \frac{1}{\color{blue}{e^{b} + 1}} \]

      2. flip-+ 0.7%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{b} \cdot e^{b} - 1 \cdot 1}{e^{b} - 1}}} \]

      3. pow2 0.7%

         \[\leadsto \frac{1}{\frac{\color{blue}{{\left(e^{b}\right)}^{2}} - 1 \cdot 1}{e^{b} - 1}} \]

      4. metadata-eval 0.7%

         \[\leadsto \frac{1}{\frac{{\left(e^{b}\right)}^{2} - \color{blue}{1}}{e^{b} - 1}} \]

      5. expm1-undefine 0.9%

         \[\leadsto \frac{1}{\frac{{\left(e^{b}\right)}^{2} - 1}{\color{blue}{\mathsf{expm1}\left(b\right)}}} \]

   5. Applied egg-rr 0.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{{\left(e^{b}\right)}^{2} - 1}{\mathsf{expm1}\left(b\right)}}} \]
   6. Step-by-step derivation

      1. unpow2 0.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{b} \cdot e^{b}} - 1}{\mathsf{expm1}\left(b\right)}} \]

      2. prod-exp 0.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{b + b}} - 1}{\mathsf{expm1}\left(b\right)}} \]

      3. expm1-define 3.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{expm1}\left(b + b\right)}}{\mathsf{expm1}\left(b\right)}} \]

   7. Simplified 3.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 3.6%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}} \cdot \sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}}} \]

      2. sqrt-unprod 3.6%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)} \cdot \frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}}} \]

      3. pow2 3.6%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\left(\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}}} \]

      4. flip-+ 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\color{blue}{\frac{b \cdot b - b \cdot b}{b - b}}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

      5. +-inverses 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{\color{blue}{0}}{b - b}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

      6. +-inverses 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{0}{\color{blue}{0}}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

   9. Applied egg-rr 0.0%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{0}{0}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}}} \]

   11. Simplified 88.5%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{e^{b + b}}}} \]

   12. Taylor expanded in b around 0 3.9%

       \[\leadsto \color{blue}{1} \]

   14. Simplified 95.9%

       \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-40}:\\ \;\;\;\;0.5 + a \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.9% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-40}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b)
 :precision binary64
 (if (<= b -2.7e-6) 1.0 (if (<= b 3e-40) 0.5 0.0)))

C

double code(double a, double b) {
	double tmp;
	if (b <= -2.7e-6) {
		tmp = 1.0;
	} else if (b <= 3e-40) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= (-2.7d-6)) then
        tmp = 1.0d0
    else if (b <= 3d-40) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= -2.7e-6) {
		tmp = 1.0;
	} else if (b <= 3e-40) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= -2.7e-6:
		tmp = 1.0
	elif b <= 3e-40:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= -2.7e-6)
		tmp = 1.0;
	elseif (b <= 3e-40)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= -2.7e-6)
		tmp = 1.0;
	elseif (b <= 3e-40)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, -2.7e-6], 1.0, If[LessEqual[b, 3e-40], 0.5, 0.0]]

Derivation

1. Split input into 3 regimes

2. if b < -2.69999999999999998e-6

   1. Initial program 95.3%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 93.6%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   4. Step-by-step derivation

      1. +-commutative 93.6%

         \[\leadsto \frac{1}{\color{blue}{e^{b} + 1}} \]

      2. flip-+ 93.4%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{b} \cdot e^{b} - 1 \cdot 1}{e^{b} - 1}}} \]

      3. pow2 93.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{{\left(e^{b}\right)}^{2}} - 1 \cdot 1}{e^{b} - 1}} \]

      4. metadata-eval 93.4%

         \[\leadsto \frac{1}{\frac{{\left(e^{b}\right)}^{2} - \color{blue}{1}}{e^{b} - 1}} \]

      5. expm1-undefine 93.4%

         \[\leadsto \frac{1}{\frac{{\left(e^{b}\right)}^{2} - 1}{\color{blue}{\mathsf{expm1}\left(b\right)}}} \]

   5. Applied egg-rr 93.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{{\left(e^{b}\right)}^{2} - 1}{\mathsf{expm1}\left(b\right)}}} \]
   6. Step-by-step derivation

      1. unpow2 93.4%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{b} \cdot e^{b}} - 1}{\mathsf{expm1}\left(b\right)}} \]

      2. prod-exp 93.5%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{b + b}} - 1}{\mathsf{expm1}\left(b\right)}} \]

      3. expm1-define 93.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{expm1}\left(b + b\right)}}{\mathsf{expm1}\left(b\right)}} \]

   7. Simplified 93.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 93.6%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}} \cdot \sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}}} \]

      2. sqrt-unprod 93.6%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)} \cdot \frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}}} \]

      3. pow2 93.6%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\left(\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}}} \]

      4. flip-+ 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\color{blue}{\frac{b \cdot b - b \cdot b}{b - b}}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

      5. +-inverses 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{\color{blue}{0}}{b - b}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

      6. +-inverses 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{0}{\color{blue}{0}}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

   9. Applied egg-rr 0.0%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{0}{0}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}}} \]

   11. Simplified 4.9%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{e^{b + b}}}} \]

   12. Taylor expanded in b around 0 91.7%

       \[\leadsto \color{blue}{1} \]

   if -2.69999999999999998e-6 < b < 3.0000000000000002e-40

   1. Initial program 100.0%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 74.5%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]

   4. Taylor expanded in b around 0 74.3%

      \[\leadsto \color{blue}{0.5} \]

   if 3.0000000000000002e-40 < b

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 91.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   4. Step-by-step derivation

      1. +-commutative 91.0%

         \[\leadsto \frac{1}{\color{blue}{e^{b} + 1}} \]

      2. flip-+ 0.7%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{b} \cdot e^{b} - 1 \cdot 1}{e^{b} - 1}}} \]

      3. pow2 0.7%

         \[\leadsto \frac{1}{\frac{\color{blue}{{\left(e^{b}\right)}^{2}} - 1 \cdot 1}{e^{b} - 1}} \]

      4. metadata-eval 0.7%

         \[\leadsto \frac{1}{\frac{{\left(e^{b}\right)}^{2} - \color{blue}{1}}{e^{b} - 1}} \]

      5. expm1-undefine 0.9%

         \[\leadsto \frac{1}{\frac{{\left(e^{b}\right)}^{2} - 1}{\color{blue}{\mathsf{expm1}\left(b\right)}}} \]

   5. Applied egg-rr 0.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{{\left(e^{b}\right)}^{2} - 1}{\mathsf{expm1}\left(b\right)}}} \]
   6. Step-by-step derivation

      1. unpow2 0.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{b} \cdot e^{b}} - 1}{\mathsf{expm1}\left(b\right)}} \]

      2. prod-exp 0.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{b + b}} - 1}{\mathsf{expm1}\left(b\right)}} \]

      3. expm1-define 3.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{expm1}\left(b + b\right)}}{\mathsf{expm1}\left(b\right)}} \]

   7. Simplified 3.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 3.6%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}} \cdot \sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}}} \]

      2. sqrt-unprod 3.6%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)} \cdot \frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}}} \]

      3. pow2 3.6%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\left(\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}}} \]

      4. flip-+ 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\color{blue}{\frac{b \cdot b - b \cdot b}{b - b}}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

      5. +-inverses 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{\color{blue}{0}}{b - b}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

      6. +-inverses 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{0}{\color{blue}{0}}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

   9. Applied egg-rr 0.0%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{0}{0}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}}} \]

   11. Simplified 88.5%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{e^{b + b}}}} \]

   12. Taylor expanded in b around 0 3.9%

       \[\leadsto \color{blue}{1} \]

   14. Simplified 95.9%

       \[\leadsto \color{blue}{0} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-40}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.0% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-40}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (if (<= b 3e-40) 0.5 0.0))

C

double code(double a, double b) {
	double tmp;
	if (b <= 3e-40) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (b <= 3d-40) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b) {
	double tmp;
	if (b <= 3e-40) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b):
	tmp = 0
	if b <= 3e-40:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b)
	tmp = 0.0
	if (b <= 3e-40)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b)
	tmp = 0.0;
	if (b <= 3e-40)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_] := If[LessEqual[b, 3e-40], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if b < 3.0000000000000002e-40

   1. Initial program 98.7%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 79.6%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]

   4. Taylor expanded in b around 0 59.4%

      \[\leadsto \color{blue}{0.5} \]

   if 3.0000000000000002e-40 < b

   1. Initial program 98.9%

      \[\frac{e^{a}}{e^{a} + e^{b}} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0 91.0%

      \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
   4. Step-by-step derivation

      1. +-commutative 91.0%

         \[\leadsto \frac{1}{\color{blue}{e^{b} + 1}} \]

      2. flip-+ 0.7%

         \[\leadsto \frac{1}{\color{blue}{\frac{e^{b} \cdot e^{b} - 1 \cdot 1}{e^{b} - 1}}} \]

      3. pow2 0.7%

         \[\leadsto \frac{1}{\frac{\color{blue}{{\left(e^{b}\right)}^{2}} - 1 \cdot 1}{e^{b} - 1}} \]

      4. metadata-eval 0.7%

         \[\leadsto \frac{1}{\frac{{\left(e^{b}\right)}^{2} - \color{blue}{1}}{e^{b} - 1}} \]

      5. expm1-undefine 0.9%

         \[\leadsto \frac{1}{\frac{{\left(e^{b}\right)}^{2} - 1}{\color{blue}{\mathsf{expm1}\left(b\right)}}} \]

   5. Applied egg-rr 0.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{{\left(e^{b}\right)}^{2} - 1}{\mathsf{expm1}\left(b\right)}}} \]
   6. Step-by-step derivation

      1. unpow2 0.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{b} \cdot e^{b}} - 1}{\mathsf{expm1}\left(b\right)}} \]

      2. prod-exp 0.9%

         \[\leadsto \frac{1}{\frac{\color{blue}{e^{b + b}} - 1}{\mathsf{expm1}\left(b\right)}} \]

      3. expm1-define 3.6%

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{expm1}\left(b + b\right)}}{\mathsf{expm1}\left(b\right)}} \]

   7. Simplified 3.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 3.6%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}} \cdot \sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}}} \]

      2. sqrt-unprod 3.6%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)} \cdot \frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}}} \]

      3. pow2 3.6%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{{\left(\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}}} \]

      4. flip-+ 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\color{blue}{\frac{b \cdot b - b \cdot b}{b - b}}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

      5. +-inverses 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{\color{blue}{0}}{b - b}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

      6. +-inverses 0.0%

         \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{0}{\color{blue}{0}}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

   9. Applied egg-rr 0.0%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{0}{0}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}}} \]

   11. Simplified 88.5%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{e^{b + b}}}} \]

   12. Taylor expanded in b around 0 3.9%

       \[\leadsto \color{blue}{1} \]

   14. Simplified 95.9%

       \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-40}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.6% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b) :precision binary64 0.0)

C

double code(double a, double b) {
	return 0.0;
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 0.0d0
end function

Java

public static double code(double a, double b) {
	return 0.0;
}

Python

def code(a, b):
	return 0.0

Julia

function code(a, b)
	return 0.0
end

MATLAB

function tmp = code(a, b)
	tmp = 0.0;
end

Wolfram

code[a_, b_] := 0.0

Derivation

1. Initial program 98.8%

   \[\frac{e^{a}}{e^{a} + e^{b}} \]
2. Add Preprocessing

3. Taylor expanded in a around 0 83.8%

   \[\leadsto \color{blue}{\frac{1}{1 + e^{b}}} \]
4. Step-by-step derivation

   1. +-commutative 83.8%

      \[\leadsto \frac{1}{\color{blue}{e^{b} + 1}} \]

   2. flip-+ 16.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{e^{b} \cdot e^{b} - 1 \cdot 1}{e^{b} - 1}}} \]

   3. pow2 16.3%

      \[\leadsto \frac{1}{\frac{\color{blue}{{\left(e^{b}\right)}^{2}} - 1 \cdot 1}{e^{b} - 1}} \]

   4. metadata-eval 16.3%

      \[\leadsto \frac{1}{\frac{{\left(e^{b}\right)}^{2} - \color{blue}{1}}{e^{b} - 1}} \]

   5. expm1-undefine 17.1%

      \[\leadsto \frac{1}{\frac{{\left(e^{b}\right)}^{2} - 1}{\color{blue}{\mathsf{expm1}\left(b\right)}}} \]

5. Applied egg-rr 17.1%

   \[\leadsto \frac{1}{\color{blue}{\frac{{\left(e^{b}\right)}^{2} - 1}{\mathsf{expm1}\left(b\right)}}} \]
6. Step-by-step derivation

   1. unpow2 17.1%

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{b} \cdot e^{b}} - 1}{\mathsf{expm1}\left(b\right)}} \]

   2. prod-exp 17.1%

      \[\leadsto \frac{1}{\frac{\color{blue}{e^{b + b}} - 1}{\mathsf{expm1}\left(b\right)}} \]

   3. expm1-define 51.4%

      \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{expm1}\left(b + b\right)}}{\mathsf{expm1}\left(b\right)}} \]

7. Simplified 51.4%

   \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 50.6%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}} \cdot \sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}}} \]

   2. sqrt-unprod 51.4%

      \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)} \cdot \frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}}}} \]

   3. pow2 51.4%

      \[\leadsto \frac{1}{\sqrt{\color{blue}{{\left(\frac{\mathsf{expm1}\left(b + b\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}}} \]

   4. flip-+ 0.0%

      \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\color{blue}{\frac{b \cdot b - b \cdot b}{b - b}}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

   5. +-inverses 0.0%

      \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{\color{blue}{0}}{b - b}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

   6. +-inverses 0.0%

      \[\leadsto \frac{1}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{0}{\color{blue}{0}}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}} \]

9. Applied egg-rr 0.0%

   \[\leadsto \frac{1}{\color{blue}{\sqrt{{\left(\frac{\mathsf{expm1}\left(\frac{0}{0}\right)}{\mathsf{expm1}\left(b\right)}\right)}^{2}}}} \]

11. Simplified 40.5%

    \[\leadsto \frac{1}{\color{blue}{\sqrt{e^{b + b}}}} \]

12. Taylor expanded in b around 0 23.7%

    \[\leadsto \color{blue}{1} \]

14. Simplified 49.3%

    \[\leadsto \color{blue}{0} \]

15. Final simplification 49.3%

    \[\leadsto 0 \]
16. Add Preprocessing

Developer target: 100.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + e^{b - a}} \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (/ 1.0 (+ 1.0 (exp (- b a)))))

C

double code(double a, double b) {
	return 1.0 / (1.0 + exp((b - a)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = 1.0d0 / (1.0d0 + exp((b - a)))
end function

Java

public static double code(double a, double b) {
	return 1.0 / (1.0 + Math.exp((b - a)));
}

Python

def code(a, b):
	return 1.0 / (1.0 + math.exp((b - a)))

Julia

function code(a, b)
	return Float64(1.0 / Float64(1.0 + exp(Float64(b - a))))
end

MATLAB

function tmp = code(a, b)
	tmp = 1.0 / (1.0 + exp((b - a)));
end

Wolfram

code[a_, b_] := N[(1.0 / N[(1.0 + N[Exp[N[(b - a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024040 
(FPCore (a b)
  :name "Quotient of sum of exps"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ 1.0 (exp (- b a))))

  (/ (exp a) (+ (exp a) (exp b))))